I don’t own the Mini, but putting sound dampening underneath and behind a PC does make it quieter in my experience. Just make sure to not block the air flow.
There are a number of companies that make low-noise CPU fans, like these:
https://premiumbuilds.com/cpu-coolers/quietest-cpu-coolers/
If you check the websites for the companies mentioned in those articles, you might be able to find one that fits inside the Librem Mini.